Development of the Theory of Additional Impact on the Deformation Zone from the Side of Rolling Rolls

The model explicitly incorporates boundary conditions that account for the complex interplay between sections experiencing varying degrees of reduction. This interaction significantly influences the overall deformation behavior and force loading. The control effect is associated with boundary conditions determined by the unevenness of the compression, which have certain quantitative and qualitative characteristics. These include additional loading, which is less than the main load, which implements the process of plastic deformation, and the ratio of control loads from the entrance and exit of the deformation site. According to this criterion, it follows from experimental data that the controlling effect on the plastic deformation site occurs with a ratio of additional and main loading in the range of 0.2–0.8. The next criterion is the coefficient of support, which determines the area of asymmetry of the force load and is in the range of 2.00–4.155. Furthermore, the criterion of the regulating force ratio at the boundaries of the deformation center forming a longitudinal plastic shear is within the limits of 2.2–2.5 forces and 1.3–1.4 moments of these forces. In this state, stresses and deformations of the plastic medium are able to realize the effects of plastic shaping. The force effect reduces with an increase in the unevenness of the deformation. This is due to a change in height of the longitudinal interaction of the disparate sections of the strip. There is an appearance of a new quality of loading—longitudinal plastic shear along the deformation site. The unbalanced additional force action at the entrance of the deformation source is balanced by the force source of deformation, determined by the appearance of a functional shift in the model of the stress state of the metal. The developed theory, using the generalized method of an argument of functions of a complex variable, allows us to characterize the functional shift in the deformation site using invariant Cauchy–Riemann relations and Laplace differential equations. Furthermore, the model allows for the investigation of material properties such as the yield strength and strain hardening, influencing the size and characteristics of the identified limit state zone. Future research will focus on extending the model to incorporate more complex material behaviors, including viscoelastic effects, and to account for dynamic loading conditions, more accurately reflecting real-world milling processes. The detailed understanding gained from this model offers significant potential for optimizing mill roll designs and processes for enhanced efficiency and reduced energy consumption.